..I've done math problems on some paper, and there are no solutions given.
Now, what if you've "gained" some (and which?) solution by squaring the original?
Mr F says: You have to go back to the original expression and test each potential solution. Clearly therefore discard the positive solution.
Again, what if you've "gained" some solution by squaring the original? What are considered best practises in solving such?
Mr F says: You have to test each potential solution in the original equation. Both solutions satisfy the original equation so you keep both.